At the HS level I always start with a discussion of very familiar symbols: $, %,
& !, ?, the peace symbol and the smiley face. I ask the students what these
symbols mean to them, and universally they all agree. Then I talk about how in
math, we need symbols as well to abbreviate meaning that eventually become a
convention, just like the dollar sign is an accepted convention....

Then I ask what a number is: a sum of powers of ten. First I write out a long
sum, like 1234.5678 in fraction form, then decimal form... then I introduce some
drama and disgust in writing so many numbers, deliberately making mistakes with
my writing of the right number of zeros, then I introduce a make-pretend
fashion of tracking those zeros and I use a negative and positive exponent with
particular emphasis that the numbers are decreasing consecutive from left to
right.... which helps with the zero exponent....

With regard to the properties, I do a detailed review of the commutative and
associative properties of multiplication and that any number, except zero,
divided by itself is always one.

On Oct 1 2004, PWR wrote:
> I always have trouble teaching exponents. To me, the rules are both
> logical and easy, but my students usually seem to find them too
> numerous and confusing. Does anyone have a gimmic, an idea like a
> mnemonic or anything at all, that helps students with problems
> involving addition, subtraction, multiplication, division, power of
> a power, negative, and fractional exponents?

This material is based upon work supported by the
National Science Foundation under Grant DUE-0226284.
Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s)
and do not necessarily reflect the views of the
National Science Foundation.